Let $Z_1, Z_2, Z_3, ...$ be a sequence of i.i.d random variables valued in $\{-1, 1\}$ and taking each value with equal probability.
Define $X_n=\sum_{i=1}^n Z_i$ and $Y_n=X_n^2-n$. It can be easily checked that the sequence $Y_1, Y_2, Y_3, \ldots$ is a martingale with respect to $X_1, X_2, X_3, \ldots$.
The third example in the section Exmaples of Martingales on this wikipedia page says that using this martingale one can say that $X_n$ varies roughly between $\sqrt n$ and $-\sqrt n$.
I can use the law of large numbers to get $P[|X_n|>n\varepsilon]\leq 2e^{-n\varepsilon^2/2}$, and put $\varepsilon = 1/\sqrt n$, but I am not able to see how the martingale $(Y_1, Y_2, Y_3, \ldots)$ helps.
Can somebody spell this out. Also, if ppossible, can you please comment as to what prompts one to consider the random variables $Y_n$.